Question

t is the midpoint of \\( \overline{ux} \\) and w is the midpoint of \\( \overline{vx} \\). if \\( uv = 6y \\) and \\( tw = - 2y + 5 \\), what is the value of y?

Explanation:

Step1: Use the mid - segment theorem

In a triangle, the segment connecting the midpoints of two sides is parallel to the third side and half its length. Since \(T\) is the midpoint of \(UX\) and \(W\) is the midpoint of \(VX\), then \(TW=\frac{1}{2}UV\).

Step2: Substitute the given expressions

We know that \(UV = 6y\) and \(TW=-2y + 5\). Substituting into the equation \(TW=\frac{1}{2}UV\), we get \(-2y + 5=\frac{1}{2}(6y)\).

Step3: Simplify the right - hand side

\(\frac{1}{2}(6y)=3y\). So the equation becomes \(-2y + 5=3y\).

Step4: Solve for \(y\)

Add \(2y\) to both sides of the equation: \(-2y+2y + 5=3y+2y\). This simplifies to \(5 = 5y\). Then divide both sides by \(5\): \(y = 1\).

Answer:

\(y = 1\)